Abstract

A new type of instability—electrokinetic instability—and an unusual transition to chaotic motion near a charge-selective surface (semiselective electric membrane, electrode, or system of micro-/nanochannels) was studied by the numerical integration of the Nernst-Planck-Poisson-Stokes system and a weakly nonlinear analysis near the threshold of instability. A special finite-difference method was used for the space discretization along with a semi-implicit -step Runge-Kutta scheme for the integration in time. Two kinds of initial conditions were considered: (a) white-noise initial conditions to mimic “room disturbances” and subsequent natural evolution of the solution, and (b) an artificial monochromatic ion distribution with a fixed wave number to simulate regular wave patterns. The results were studied from the viewpoint of hydrodynamic stability and bifurcation theory. The threshold of electroconvective movement was found by the linear spectral stability theory, the results of which were confirmed by numerical simulation of the entire system. Our weakly nonlinear analysis and numerical integration of the entire system predict possibility of both kinds of bifurcations at the critical point, supercritical and subcritical, depending on the system parameters. The following regimes, which replace each other as the potential drop between the selective surfaces increases, were obtained: one-dimensional steady solution, two-dimensional steady electroconvective vortices (stationary point in a proper phase space), unsteady vortices aperiodically changing their parameters (homoclinic contour), periodic motion (limit cycle), and chaotic motion. The transition to chaotic motion does not include Hopf bifurcation. The numerical resolution of the thin concentration polarization layer showed spike-like charge profiles along the surface, which could be, depending on the regime, either steady or aperiodically coalescent. The numerical investigation confirmed the experimentally observed absence of regular (near-sinusoidal) oscillations for the overlimiting regimes. There is a qualitative agreement of the experimental and the theoretical values of the threshold of instability, the dominant size of the observed coherent structures, and the experimental and theoretical volt–current characteristics.

Received 14 June 2013Accepted 22 November 2013Published online 17 December 2013

Acknowledgments:

This work was supported, in part, by the Russian Foundation for Basic Research (Project Nos. 11-01-00088-a, 11-01-96505-r_yug_ts, 11-08-00480-a, and 12-08-00924-a).

Article outline:I. INTRODUCTIONA. General backgroundB. Theoretical foundations of electrokinetic instabilityC. Numerical tools and methods of the present workII. THE MATHEMATICAL MODELIII. THE NUMERICAL METHODA. Discretization of the Nernst-Planck-Poisson-Stokes systemB. Domain length and discretization of the initial conditionsIV. SIMULATION RESULTSA. 1D trivial solution and its stabilityB. Weakly nonlinear analysis and behavior near thresholdC. Volt-current characteristic and hysteresisD. Family of 2D steady space-periodic solutions and their stabilityE. Homoclinic contours, their bifurcations, and the birth of chaosF. Spatial spectraG. Comparison with experimentsV. CONCLUSIONS